Zero-determinant (ZD) strategies are a novel class of strategies in the repeated prisoner's dilemma (RPD) game discovered by Press and Dyson. This strategy set enforces a linear payoff relationship between a focal player and the opponent regardless of the opponent's strategy. In the RPD game, games with discounting and observation errors represent an important generalization, because they are better able to capture real life interactions which are often noisy. However, they have not been considered in the original discovery of ZD strategies. In some preceding studies, each of them has been considered independently. Here, we analytically study the strategies that enforce linear payoff relationships in the RPD game considering both a discount factor and observation errors. As a result, we first reveal that the payoffs of two players can be represented by the form of determinants as shown by Press and Dyson even with the two factors. Then, we search for all possible strategies that enforce linear payoff relationships and find that both ZD strategies and unconditional strategies are the only strategy sets to satisfy the condition. We also show that neither Extortion nor Generous strategies, which are subsets of ZD strategies, exist when there are errors. Finally, we numerically derive the threshold values above which the subsets of ZD strategies exist. These results contribute to a deep understanding of ZD strategies in society.

中文翻译：

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重复游戏中观察误差下的零决定策略

零决定因素（ZD）策略是Press and Dyson发现的重复囚徒困境（RPD）游戏中的一类新颖策略。无论对手的策略如何，此策略集都会在焦点玩家和对手之间强制执行线性收益关系。在RPD游戏中，具有折扣和观察误差的游戏代表了一种重要的概括，因为它们能够更好地捕获经常嘈杂的现实生活中的互动。但是，ZD策略的原始发现中尚未考虑它们。在先前的一些研究中，每个都被独立考虑。在这里，我们分析性地研究了在考虑折扣系数和观察误差的情况下在RPD游戏中实施线性收益关系的策略。结果是，我们首先揭示出，即使有两个因素，也可以用Press和Dyson所示的行列式表示两个参与者的收益。然后，我们搜索所有强制执行线性收益关系的策略，并发现ZD策略和无条件策略都是满足条件的唯一策略集。我们还表明，存在错误时，作为ZD策略的子集的勒索策略和慷慨策略都不存在。最后，我们从数值上得出阈值，高于该阈值则存在ZD策略的子集。这些结果有助于深入了解ZD策略在社会中。我们搜索了强制执行线性收益关系的所有可能策略，发现ZD策略和无条件策略都是满足条件的唯一策略集。我们还表明，存在错误时，作为ZD策略的子集的勒索策略和慷慨策略都不存在。最后，我们从数值上得出阈值，高于该阈值则存在ZD策略的子集。这些结果有助于深入了解ZD策略在社会中。我们搜索了强制执行线性收益关系的所有可能策略，发现ZD策略和无条件策略都是满足条件的唯一策略集。我们还表明，存在错误时，作为ZD策略的子集的勒索策略和慷慨策略都不存在。最后，我们从数值上得出阈值，高于该阈值则存在ZD策略的子集。这些结果有助于深入了解ZD策略在社会中。